Question

Use long division to divide the polynomial 15x2+14x−8 by 3x+4. What is the quotient?(1 point)
Responses

5x+2
5 x plus 2

5x+2−163x+4
5 x plus 2 minus Start Fraction 16 over 3 x plus 4 End Fraction

5x−2
5 x minus 2

5x−2+163x+4
5 x minus 2 plus Start Fraction 16 over 3 x plus 4 End Fraction

Answer 1

To divide the polynomial \( 15x^2 + 14x - 8 \) by \( 3x + 4 \) using long division, follow these steps:

1. **Set up the long division**: Write \( 15x^2 + 14x - 8 \) under the long division bar and \( 3x + 4 \) outside.

2. **Divide the leading term**: Divide the leading term of the dividend \( 15x^2 \) by the leading term of the divisor \( 3x \):
\[
\frac{15x^2}{3x} = 5x
\]

3. **Multiply**: Multiply \( 5x \) by the entire divisor \( 3x + 4 \):
\[
5x(3x + 4) = 15x^2 + 20x
\]

4. **Subtract**: Subtract this result from the original polynomial:
\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\]

5. **Bring down**: There is nothing to bring down in this case.

6. **Repeat**: Now divide the leading term \(-6x\) by \(3x\):
\[
\frac{-6x}{3x} = -2
\]

7. **Multiply**: Multiply \(-2\) by the divisor:
\[
-2(3x + 4) = -6x - 8
\]

8. **Subtract**: Subtract again:
\[
(-6x - 8) - (-6x - 8) = 0
\]

At this point, there is no remainder. Thus, the total quotient of the division is:
\[
\boxed{5x - 2}
\]

So the final answer is:
5x−2